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شائع علم المثلثات >

solvefor x,3sec^4(x)-10sec^2(x)+8=0

الحلّ

solve for

x, 3 sec^{4} (x) - 10 sec^{2} (x) + 8 = 0

الحلّ

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

درجات

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

3 sec^{4} (x) - 10 sec^{2} (x) + 8 = 0

بالاستعانة بطريقة التعويض

3 sec^{4} (x) - 10 sec^{2} (x) + 8 = 0

sec (x) = u :

على افتراض أنّ

3 u^{4} - 10 u^{2} + 8 = 0

3 u^{4} - 10 u^{2} + 8 = 0 : u = 2, u = - 2, u = \frac{2 3}{3}, u = - \frac{2 3}{3}

3 u^{4} - 10 u^{2} + 8 = 0

v^{2} = u^{4}

وكذلك

v = u^{2}

اكتب المعادلة مجددًا، بحيث أنّ

3 v^{2} - 10 v + 8 = 0

3 v^{2} - 10 v + 8 = 0

حلّ:

v = 2, v = \frac{4}{3}

3 v^{2} - 10 v + 8 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

3 v^{2} - 10 v + 8 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = 3, b = - 10, c = 8

لـ

v_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 3 \cdot 8}{2 \cdot 3}

v_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 3 \cdot 8}{2 \cdot 3}

(- 10)^{2} - 4 \cdot 3 \cdot 8 = 2

(- 10)^{2} - 4 \cdot 3 \cdot 8

زوجيّ n

إذا تحقّق أنّ

, (- a)^{n} = a^{n}

:فعّل قانون القوى

(- 10)^{2} = 1 0^{2}

= 1 0^{2} - 4 \cdot 3 \cdot 8

4 \cdot 3 \cdot 8 = 96 :

اضرب الأعداد

= 1 0^{2} - 96

1 0^{2} = 100

= 100 - 96

100 - 96 = 4 :

اطرح الأعداد

= 4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

v_{1, 2} = \frac{- ( - 10 ) \pm 2}{2 \cdot 3}

Separate the solutions

v_{1} = \frac{- ( - 10 ) + 2}{2 \cdot 3}, v_{2} = \frac{- ( - 10 ) - 2}{2 \cdot 3}

v = \frac{- ( - 10 ) + 2}{2 \cdot 3} : 2

\frac{- ( - 10 ) + 2}{2 \cdot 3}

- (- a) = a

فعّل القانون

= \frac{10 + 2}{2 \cdot 3}

10 + 2 = 12 :

اجمع الأعداد

= \frac{12}{2 \cdot 3}

2 \cdot 3 = 6 :

اضرب الأعداد

= \frac{12}{6}

\frac{12}{6} = 2 :

اقسم الأعداد

= 2

v = \frac{- ( - 10 ) - 2}{2 \cdot 3} : \frac{4}{3}

\frac{- ( - 10 ) - 2}{2 \cdot 3}

- (- a) = a

فعّل القانون

= \frac{10 - 2}{2 \cdot 3}

10 - 2 = 8 :

اطرح الأعداد

= \frac{8}{2 \cdot 3}

2 \cdot 3 = 6 :

اضرب الأعداد

= \frac{8}{6}

2 :

إلغ العوامل المشتركة

= \frac{4}{3}

حلول المعادلة التربيعيّة هي

v = 2, v = \frac{4}{3}

v = 2, v = \frac{4}{3}

Substitute back

v = u^{2},

solve for

u

u^{2} = 2

حلّ:

u = 2, u = - 2

u^{2} = 2

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = 2, u = - 2

u^{2} = \frac{4}{3}

حلّ:

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

u^{2} = \frac{4}{3}

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = \frac{4}{3}, u = - \frac{4}{3}

\frac{4}{3} = \frac{2 3}{3}

\frac{4}{3}

a \geq 0, b \geq 0

بافتراض أنّ

n \frac{a}{b} = \frac{n a}{n b}

:فعّل قانون الجذور

= \frac{4}{3}

4 = 2

4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

= \frac{2}{3}

\frac{2}{3}

حوّل لصيغة عدد كسريّ:

\frac{2 3}{3}

\frac{2}{3}

\frac{3}{3}

اضرب بالمرافق

= \frac{2 3}{3 3}

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

- \frac{4}{3} = - \frac{2 3}{3}

- \frac{4}{3}

\frac{4}{3}

بسّط:

\frac{2}{3}

\frac{4}{3}

a \geq 0, b \geq 0

بافتراض أنّ

n \frac{a}{b} = \frac{n a}{n b}

:فعّل قانون الجذور

= \frac{4}{3}

4 = 2

4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

= \frac{2}{3}

= - \frac{2}{3}

- \frac{2}{3}

حوّل لصيغة عدد كسريّ:

- \frac{2 3}{3}

- \frac{2}{3}

\frac{3}{3}

اضرب بالمرافق

= - \frac{2 3}{3 3}

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

The solutions are

u = 2, u = - 2, u = \frac{2 3}{3}, u = - \frac{2 3}{3}

u = sec (x)

استبدل مجددًا

sec (x) = 2, sec (x) = - 2, sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}

sec (x) = 2, sec (x) = - 2, sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = 2

sec (x) = 2 :

حلول عامّة لـ

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) = - 2

sec (x) = - 2 :

حلول عامّة لـ

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) = \frac{2 3}{3} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) = \frac{2 3}{3}

sec (x) = \frac{2 3}{3} :

حلول عامّة لـ

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) = - \frac{2 3}{3} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

sec (x) = - \frac{2 3}{3}

sec (x) = - \frac{2 3}{3} :

حلول عامّة لـ

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

وحّد الحلول

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

رسم

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